513.001 Molecular and Solid State Physics

Ionized donors

Silicon is doped with phosphorus at 4 × 10¹⁶ 1/cm³. At what temperature are 30% of the donors ionized?

This is not an easy question. If we knew the Fermi energy, the fraction of ionized donors could be calculated using the formula,

Since the Fermi energy is not given, it will have to be determined numerically. The Fermi energy can be determined from the charge neutrality condition,

n = N_d⁺ + p.

One way to proceed is to program the formulas for n, N_d⁺, and p in a spreadsheet. Then choose a temperature and calculate n, N_d⁺, and p for every value of the Fermi energy between E_v and E_c. For one of these E_f values, the charge neutrality condition will be satisfied. The fraction of ionized donors at that temperature can then be read off for the Fermi energy which satisfies the charge neutrality condition. This procedure would have to be repeated for different temperatures.